Takahashi, Hiroshi; Kanagawa, Shuya; Yoshihara, Ken-Ichi Asymptotic behavior of solutions of some difference equations defined by weakly dependent random vectors. (English) Zbl 1333.39014 Stochastic Anal. Appl. 33, No. 4, 740-755 (2015). The authors are motivated by the recent paper of K.-I. Yoshihara [Yokohama Math. J. 58, 1–15 (2012; Zbl 1278.60099)], where the relationship between the Black-Scholes-type differential equation \(dX(t)=\sigma X(t)\,dW(t)+\nu X(t)\,dt\) and its discrete counterpart \[ \Delta X(t_k)=X(t_k)-X(t_{k-1})=X(t_{k-1})\left[\frac{\nu T}{n}+ \sqrt{\frac{T}{n}}\sigma \xi_k\right] \] is investigated. The present paper extends this result in two directions: (i) a pair of more general equations \[ dX(t)=\,X(t)\left[h(t)dt+v(t)\,dW(t)\right], \] and \[ \Delta X(t_k)=\,X(t_{k-1})\left[h(t_{k-1})\frac{T}{n}+ \sqrt{\frac{T}{n}}v(t_{k-1})\xi_k\right] \tag{*} \] is considered, (ii) the quantities \(X,\xi\) are allowed to be multidimensional.Additional conditions are formulated under which the quantity \(S^{(n)}(T)\) for the solution \[ S^{(n)}(t)=z\exp\left\{\sum_{k=1}^n \log\left(1+h(t_{k-1}) \frac{T}{n}+\sqrt{\frac{T}{n}}v(t_{k-1})\xi_k\right)\right\} \] of (*) converges almost surely to \[ x(T)=z\exp\left\{\int_0^T h(t)\,dt-\frac{1}{2}\int_0^T v^2(t)\,dt+\gamma_1 \int_0^T v(t)\,dW(t)\right\}. \] Some extension of this result to a more general pair of equations is also formulated. Reviewer: Ondřej Došlý (Brno) Cited in 1 ReviewCited in 3 Documents MSC: 39A50 Stochastic difference equations 39A30 Stability theory for difference equations 60G10 Stationary stochastic processes 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 39A12 Discrete version of topics in analysis Keywords:difference equation; weakly dependent random variables; Euler-Maruyama scheme; Black-Scholes-type stochastic differential equation Citations:Zbl 1278.60099 PDFBibTeX XMLCite \textit{H. Takahashi} et al., Stochastic Anal. Appl. 33, No. 4, 740--755 (2015; Zbl 1333.39014) Full Text: DOI References: [1] Kanagawa S., Yokohama Mathematical Journal 36 (1) pp 79– (1988) [2] Kanagawa S., Sugaku Expositions 18 (1) pp 75– (2005) [3] DOI: 10.1007/978-3-662-12616-5 · doi:10.1007/978-3-662-12616-5 [4] Schurz H., Handbook of Stochastic Analysis and Applications pp 237– (2002) [5] DOI: 10.1214/154957805100000104 · Zbl 1189.60077 · doi:10.1214/154957805100000104 [6] DOI: 10.1007/BF01025872 · Zbl 0451.60027 · doi:10.1007/BF01025872 [7] DOI: 10.1214/105051606000000565 · Zbl 1121.60033 · doi:10.1214/105051606000000565 [8] Yoshihara K., Yokohama Mathematical Journal 58 pp 1– (2012) [9] DOI: 10.1007/BF00533093 · Zbl 0308.60029 · doi:10.1007/BF00533093 [10] DOI: 10.1007/BF00532688 · Zbl 0307.60045 · doi:10.1007/BF00532688 [11] DOI: 10.1214/aop/1176994565 · Zbl 0451.60008 · doi:10.1214/aop/1176994565 [12] DOI: 10.1214/aop/1176995146 · Zbl 0392.60024 · doi:10.1214/aop/1176995146 [13] Yoshihara K., Weakly Dependent Stochastic Sequences and Their Applications 33 (1992) · Zbl 0922.60007 [14] DOI: 10.1016/j.spa.2008.01.012 · Zbl 1157.60311 · doi:10.1016/j.spa.2008.01.012 [15] DOI: 10.1214/10-AOP603 · Zbl 1236.60037 · doi:10.1214/10-AOP603 [16] Zaitsev A.Yu, Proceedings of the International Congress of Mathematicians, vol. III (Beijing, 2002) pp 107– (2002) [17] DOI: 10.1007/s10958-009-9682-x · Zbl 1288.60039 · doi:10.1007/s10958-009-9682-x [18] DOI: 10.5705/ss.2008.223 · Zbl 1251.60029 · doi:10.5705/ss.2008.223 [19] DOI: 10.1007/BF00534186 · Zbl 0407.60002 · doi:10.1007/BF00534186 [20] DOI: 10.1007/978-1-4615-8065-2 · doi:10.1007/978-1-4615-8065-2 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.